An example is provided by the local system \({\mathbb {V}}_{11,3,3}\)  whose fibre for a curve \(C\) 
is the irreducible representation of highest weight \((11,3,3)\)  on
\(H^1_{\rm et}(C,{\mathbb {Q}}_{\ell })\) , see [1]}.
We see a 6-dimensional motive of weight 23
appearing in the cohomology
with Hodge degrees \(0,5,9,14,18,23\) .
It cannot come
from a Siegel modular form of degree 3. It should correspond to a
Teichmüller modular form of weight \((8,0,7)\) , that is, a section of
\({\rm Sym}^8({\mathbb {E}}) \otimes {\det }({\mathbb {E}})^{\otimes 7}\)  on \(\overline{\mathcal {M}}_{3}\) .
We can construct such a Teichmüller modular form. It is fascinating to be able to
explore this cohomology and these motives simply by counting points on curves
over finite fields.
